{Magå‘V ¶m§{ÌH$s Ed§ gm§p»¶H$s ^m¡{VH$s Paper - MPH-01

MPH-01

December - Examination 2017 MSC (Previous) Physics Examination Classical Mechanics and Statistical Physics

{Magå‘V ¶m§{ÌH$s Ed§ gm§p»¶H$s ^m¡{VH$s

Paper - MPH-01

Time : 3 Hours ] [ Max. Marks :- 80

Note: The question paper is divided into three sections A, B and C. Write answers as per the given instructions, Check your paper code and paper title before starting the paper. In case of any discrepancy English version will be final for all purposes, For paper MPH-01 calculators are not allowed.

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^MPH-01

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Section - A 8

×

2 = 16

(Very Short Answer Type Questions) (compulsory)

Note: Answer all questions as per the nature of the question. Delimit your answer in one word, one sentence of or maximum upto 30 words. Each question carries 02 marks.

780

MPH-01 / 800 / 6 (1) (P.T.O.)

MPH-01 / 800 / 6 (2) (Contd.)

IÊS> - "A'

(A{V bKw CÎma dmbo àíZ) (A{Zdm¶©)

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1) (i) Write Hamilton’s principle of least action.

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(ii) What are Integrals of motion?

J{V Ho$ g‘mH$bH$ ³¶m h¢?

(iii) Write the lagrangian of a system of two interacting particle.

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(iv) What do you mean by the laboratory system and the centre of mass system in the body scattering problem.

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(v) Define Intertial tensors. Write expression for I₁₁, I₂₂ and I₃₃.

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^I11, I₂₂

Am¡a

^I33

Ho$ 춧OH$> {b{I¶o&

(vi) Write difference between Macroscopic and Microscopic parameter.

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(vii) Define grand canonical ensemble.

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MPH-01 / 800 / 6 (3) (P.T.O.)

(viii) Write the Maxwellian distribution for velocities in spherical coordinates.

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Section - B 4

×

8 = 32

(Short Answer Type Questions)

Note: Answer any four questions. Each answer should not exceed 200 words. Each question carries 8 marks.

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2) Find the equation of motion of one dimensional harmonic oscillator using Hamilton’s principle.

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3) How does the Lagrangian dxdt

1

L= -b l² transform when we change to the coordinate q and

“time” τ through the equations.

x = q coshλ + τ sinhλ t = q sinhλ + τ coshλ ?

dxdt 1

L= -b l²

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^q

VWm gԦ

^τ

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x = q coshλ + τ sinhλ t = q sinhλ + τ coshλ

MPH-01 / 800 / 6 (4)

4) Determine the oscillations of system with two degrees of freedom

whose Lagrangian is 12 x y x y xy

12

L^{= ` -2+ -2j-}

w

₀^{2` 2+ 2j+a}

(two identical one dimensional systems of eigenfrequency ω₀ coupled by an interaction – αxy

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boJaopÝO¶Z$ H$m ‘mZ {ZåZ àH$ma h¡&

x y x y xy

12

12

L^{= ` -2+ -2j-}

w

₀^{2` 2+ 2j+a}

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^ω0

h¡ VWm do

^{– αxy}

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5) What is legendre transformation? Using Legendre transformation obtain Hamilton’s equation of motion from Langranges equation of motion.

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6) Why are factors 1

N and 1 h^3N introduced into the derivation of partition function of the ideal classical gas?

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VWm

^{1 h3N}

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7) What are physical significance of the various statistical quantities in the canonical ensemble.

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MPH-01 / 800 / 6 (5)

8) Prove that the total radiation energy is proportional to the fourth power of temperature.

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9) Find the Maxwellian distribution for velocities in cylindrical coordinates in velocity space.

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Section - C 2

×

16 = 32 (Long Answer Questions)

Note: Answer any two questions. You have to delimit your each answer maximum up to 500 words. Each question carries 16 marks.

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10) Determine the period of oscillations of a simple pendulum as a function of the amplitude of oscillations.

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11) Find the principal moments of inertia of a hollow sphere about diameter.

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H$s{OE&

MPH-01 / 800 / 6 (6)

12) Using poisson's bracket check that the transformation whether canonical or not canonical.

, cos ( )

e p pe

Q=` ^-2q- ^{2 2}j¹ P= ^{-1 q}

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, cos ( )

e p pe

Q=` ^-2q- ^{2 2}j¹ P= ^{-1 q}

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13) Find the entropy S(E, V, N) of an ideal gas of N classical monoatomic particles, with a fixed total energy E, combined in a d-dimensional box of volume V.

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^N

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H$s EoÝQ´>mo{n

S(E, V, N) d

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H$s{OE&